| L(s) = 1 | + 0.863·5-s − i·7-s − 2.62i·11-s + 1.01i·13-s + 2.34i·17-s + 4.09·19-s − 6.23·23-s − 4.25·25-s − 1.57·29-s + 6.29i·31-s − 0.863i·35-s + 5.18i·37-s + 10.1i·41-s + 0.496·43-s + 5.24·47-s + ⋯ |
| L(s) = 1 | + 0.386·5-s − 0.377i·7-s − 0.792i·11-s + 0.282i·13-s + 0.568i·17-s + 0.939·19-s − 1.29·23-s − 0.850·25-s − 0.292·29-s + 1.13i·31-s − 0.145i·35-s + 0.851i·37-s + 1.58i·41-s + 0.0756·43-s + 0.764·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.777017086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.777017086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 0.863T + 5T^{2} \) |
| 11 | \( 1 + 2.62iT - 11T^{2} \) |
| 13 | \( 1 - 1.01iT - 13T^{2} \) |
| 17 | \( 1 - 2.34iT - 17T^{2} \) |
| 19 | \( 1 - 4.09T + 19T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 + 1.57T + 29T^{2} \) |
| 31 | \( 1 - 6.29iT - 31T^{2} \) |
| 37 | \( 1 - 5.18iT - 37T^{2} \) |
| 41 | \( 1 - 10.1iT - 41T^{2} \) |
| 43 | \( 1 - 0.496T + 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 4.40iT - 59T^{2} \) |
| 61 | \( 1 - 9.59iT - 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 2.11T + 71T^{2} \) |
| 73 | \( 1 + 8.27T + 73T^{2} \) |
| 79 | \( 1 + 9.44iT - 79T^{2} \) |
| 83 | \( 1 + 6.62iT - 83T^{2} \) |
| 89 | \( 1 - 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.171880654502782434873991883791, −7.51436981589557301615314904122, −6.70950013810355448702255711563, −5.97150798395764124879442024340, −5.49077671873469782056080246775, −4.47714020174526426409491514365, −3.74560018851137682617249887196, −2.97019361715637716924020912573, −1.90402957536163767822919006187, −0.985446495321195657642635452184,
0.50493093264385298419005431003, 1.95698808706350335899055524786, 2.40491690932448481142113431272, 3.62069621996836836485742398539, 4.25096771107005361097664768147, 5.39452907317464475569045203361, 5.61394715305397820163782311334, 6.55288102826981611813942273197, 7.41086911614151389499606566166, 7.80268577880323732054664820597
